Question: Let $h$ be a vector-valued function defined by $h(t)=(-t^5+3t^4,-5^{t+2})$. Find $h$ 's second derivative $h''(t)$. Choose 1 answer: Choose 1 answer: (Choice A) A $\left(-5t^4+12t^3,-5^{t+2}\ln(5)\right)$ (Choice B) B $\left(-15t^3+12t^2,-5^{t+2}\right)$ (Choice C) C $\left(-20t^3+36t^2,-5^{t+2}(\ln(5))^2\right)$ (Choice D) D $-5t^4+12t^3-5^{t+2}\ln(5)$
Solution: We are asked to find the second derivative of $h$. This means we need to differentiate $h$ twice. In other words, we differentiate $h$ once to find $h'$, and then differentiate $h'$ (which is a vector-valued function as well) to find $h''$. Recall that $h(t)=(-t^5+3t^4,-5^{t+2})$. Therefore, $h'(t)=\left(-5t^4+12t^3,-5^{t+2}\ln(5)\right)$. Now let's differentiate $h'(t)=\left(-5t^4+12t^3,-5^{t+2}\ln(5)\right)$ to find $h''$. $h''(t)=\left(-20t^3+36t^2,-5^{t+2}(\ln(5))^2\right)$ In conclusion, $h''(t)=\left(-20t^3+36t^2,-5^{t+2}(\ln(5))^2\right)$.